I just got out the sewers with one of my Oblivon characters, an Argonian named Axeman, what I'm doing with him is quite challenging:
Here's the rules, bet you can't follow them to level 20 and first start the mainquest and complete it by then:

The only spells you may cast are greater powers and the spells from your skill (except for soultrap you are allowed to get that).

You must be born under the serpeant.

Your mayor skills must be: Acrobatics, Alchemy, Block, Weapon skill of choice (blade, bow, blunt), A magic skill of choice, sneak, and security. Your favored attributes must be streangth and luck. You can chose whatever specialisation you want.

You cannot fast-travel.

You may not change the difficulty bellow or above medium.

You can enchant and make/use potions (never sell 'em) freely or use pre-enchanted clothing, armor, and weapons.

You cannot repair weapons/armor nor wear armor with 0 armor points... (But you can take it to a smith and get it repaired)

Here's some advice:

Use alchemy as a poison not potion...

Make much use of sneak.

What do you think?

Sorry for bad grammer my spelling check is messed up.

I would pick up my usual 6 companions and total blow everything away like I usually do.

I think its do-able, using Morag Tong Armor and the bladeturn hood.
Can you enchant with spells outside of the your chosen major? because that would make a big difference.
Biggest difficulty I see coming is that the axe is going to constantly be breaking, necessitating carrying several different ones.
We can use shields right?
We don't have to be an argonian do we?

Not trying it out.
But it doesn't sound too hard. I think the only thing I'd struggle with is knowing that I could get stuff so much better so very easily.

Doesn't have to be an Argonian...

I'm not going to try this. While I like significantly gimped characters, the nature of the character you propose is not my goblet of Tamika's.

I can't really offer much of an opinion on the second half of your poll question regarding difficulty. I think much of it depends on the experience of the player. I suppose sneak + alchemy + axe + illusion could allow a pretty successful experience.

I don't really think in terms of difficulty or 'beating the game'. Good luck though!

An interesting idea, yes. But not really my cup of Kahve.

I'm like Acadian, I've taken to gimping my characters but I'm too into my own storylines and suchlike to try something like this.

Good luck though, it could be a challenge.

Changed it a bit... making it an overall better thing...